2906
The Journal of Physical Chemistry, Vol. 83, No. 22, 1979
e
+
C
:
Q
-L H H
1 pam
t13ppm
Flgure 5. Schematic illustration of the effect of the second phenyl group of DPM on a typical ring current shift.
Tyrreli
forming a 1:l complex but more or less like a single benzene when it forms a 1:2 complex. The details of the geometries of the average 1:l and 1:2 complexes are too difficult to determine in all three systems, but the whole picture seems to point consistently to the relevance of 1:2 complexes in which the two molecules of donor are situated on different sides with respect to the main plane of the acceptor molecule. It is not inconceivable that even higher order complexations play a role in very concentrated donor solutions. In any case it seems in order to collect more data at high enough fields before it may be possible to go back to detailed geometrical pictures, because most of the data previously collected at 1:41 T are probably of little significance.
11
Acknowledgment. We thank the “Centro Interfacolti di Metodologie Chimico-Fisiche” of the University of Naples for the use of their WH-270 spectrometer. The skillful technical help of Miss Lucia Pastore, Universiti della Calabria, and of Mr. Raffaele Turco of L.M.I.B., is also gratefully acknowledged.
\\
References and Notes 1 2
Flgure 6. Complexation scheme proposed for the system caffeinediphenyimethane.
that it seems possible to propose a unique geometrical arrangement for the 1:l complex. Figure 6 shows this model &e., the one in which a molecule of DPM “chelates” that of caffeine by placing the two phenyl rings on opposite sides of the acceptor) together with the scheme of the facile interconversion between different 1:l complexes by internal rotation. This model explains immediately the trend of the limiting shifts; each DPM behaves as two benzenes when
(1) R. Foster, “Organic Charge Transfer Complexes”, Academic Press, New York, 1969. (2) R. Foster in “Molecular Complexes”, Voi. 2, R. Foster, Ed., Eiek Science, London, 1974. (3) J. V. Hatton and R. E. Richards, Mol. Phys., 5, 139 (1962). (4) J. Ronayne and D. H. Williams, J . Chem. SOC. B , 540 (1967). (5) R. E. Kllnck and J. 6 . Stothers, Can. J. Chem., 44, 37 (1966). (6) M. 1. Foreman, R. Foster, and D.R. Twiseiton, Chem. Commun., 1316 (1969). (7) S. Andini, L. Ferrara, P.A. Temussi, F. Lelj, and T. Tancredi, J. Phys. Chem., 83, 1766 (1979). (6) A. A. Sandoval and M. W. Hanna, J. Phys. Chem., 70, 1203 (1966). (9) M. W. Hanna and A. Sandoval, Biochim. Siophys. Acta, 155, 433 (1968). (10) B. Dodson, R. Foster, and A. A. S.Bright, J , Chetn. SOC.B, 1283 (1971). (11) R. Fletcher, Hatwell Technical Report No. R. 6799 AERE, U.K. (1971). (12) J. S. Waugh and R. W. Fessenden, J. Am. Chem. SOC.,79, 846 (1957).
Electronic Structure and One-Electron Properties of the Isoelectronic Molecules HCN, HNC, HBO, HOB, HBF’, and HFB’ James Tyrrell Department of Chemistry and 6iocbemistry, Southern Illinois University at Carbondale, Carbondale, Illlnois 6290 1 (ReC8lVed April 6, 1979)
A double quality basis set was used to determine the equilibrium geometries, energies, and one-electron properties of a number of isoelectronic molecules. The properties determined included multipole moments, electric field gradients, and diamagnetic susceptibilities. The geometries and properties were compared with experimental results and previous calculations where available.
The Journal of Physical Chemistry, Vol. 83, No. 22, 1979 2907
Electronic Structure of Isoelectronic Molecules
TABLE I: Equilibrium Geometries and Energies for the HXY Speciesa RH-X,
HCN lit. values HNC lit. values HBO lit. values HOB lit. values HBF' lit. values HFB+ lit. values
a
1.056 (1.066y 1.06314 0.981 (0.9940V 0.986li 1.154 1.179," 1.153," 1.157" 0.937 0.936,2' O.95Oa3 1.159 1.160,221.199% 0.932 0.94 5 "
RX-Y,
a
1.145 (1.153)5 1.13714 1.160 (1.1689)6 1.15914 1.202 1.177,'' 1.199," 1.20523 1.275 1.298," 1.29523 1.248 1.242,*' 1.22724 1.985 1.877"
total energy
-V/2T
-92.8427
0.99985
-92.8383
0.99954
-100.1439
0,99980
-100.0968
0.99927
-1 24.3805
1.00035
-124.3042
0.99988
a Values in parentheses are experimental results. Superscripts indicate the appropriate reference source. Energies are given in Hartrees.
or comparison with experimental results. Of the molecules being considered here only HCN and HNC have experimentally determined geometries, and experimentally measured one-electron properties are available only for HCN. The molecule, HBO, has been prepared by using matrix isolation techniques and its infrared spectrum has been ~ b t a i n e d . ~ The experimental geometry of HCN5 indicates a linear structure with an H-C bond length of 1.066 A and a C-N bond length of 1.153 A. Creswell and Robiette! using the data from the analysis of the millimeter wave spectrum of HNC by Pearson et al.,' determined the equilibrium bond lengths in HNC to be H-N = 0.9940 A and N-C = 1.1689 A. Bhattacharya and GordyBobtained a dipole moment for HCN of 2.986 D and also determined the 14N quadrupole coupling constant in H12C14Nto be -4.714 MHz. There have been a number of experimental determinations of the deuteron quadrupole coupling constant in DCN giving values of 194 f 2,g 199.0 f 3,1° and 202.18 f 0.31 kHz." Hartford et al.12 found the quadrupole moment for HC15N along the molecular axis to be 3.1 f 0.6 eswcm2. Gierke and Flygare13 used the atom dipole method to obtain the average diamagnetic shieldings u,,d('H), ua,d(13C),and aa,d(14N)in HCN to be 97,326, and 381 ppm, respectively. There is no similar experimental information available for the other molecules. Numerous ab initio calculations have been performed on HCN and on HNC with the primary purpose of obtaining equilibrium geometries. Pearson et al.14reported equilibrium geometries and dipole moments for HCN and HNC with both a double { plus polarization basis set and CI. The double {plus polarization basis set gave a H-C bond length of 1.063 A, a C-N bond length of 1.137 A, and a dipole moment of 3.22 D while for HNC it gave a H-N bond length of 0.986 A, a N-C bond length of 1.159 A, and a dipole moment of 2.91 D. The CI calculation resulted in a geometry for HCN almost identical with the experimentally determined one and a dipole moment of 3.012 D. In addition, a variety of workers using a variety of basis sets have determined particular one-electron properties for HCN. Kern and Karplus15 have calculated quadrupole coupling constants for DCN and HCN as have Bonaccorsi, Scrocco, and Tomasi.l6 Ditchfieldl' has determined magnetic shielding constants for H I T N and HC14N. Kato, Fujimoto, and Saikal* have obtained diamagnetic susceptibilities and anisotropies for HCN. Gready et al.19have determined a number of one-electron properties, including multipole moments for both HCN and HNC. Barfield and Grantmcalculated average diamagnetic shieldings a,,d(lH), aa,d(13C),and u,d(14N) for HCN in the INDO approximation, but including certain two-center integrals, to be 101.3,324.8, and 377.9 ppm, respectively. Thomson and Wishart21 used a double { plus polarization basis set on
HBO to give an equilibrium geometry (H-B = 1.179 8, B-0 = 1.177 A), a dipole moment (3.261 D), and atomcentered quadrupole moments and field gradients. Summers and Tyrre11,22using a 4-31G basis set, obtained optimized geometries for HBO (H-B = 1.153 A, B-0 = 1.199 A) and for HOB (H-0 = 0.936 A, 0-B = 1.298 A). Recently B o t s ~ h w i n ahas ~ ~ used PNO-CI and CEPA techniques to obtain optimized geometries for HBO (H-B = 1.157 A, B-0 = 1.205 A) and for HOB (H-0 = 0.950 A, 0-B = 1.295 A). Equilibrium geometries have also been calculated for HBF+ by Jansen and Rosa and by Summers and Tyrrel122and for HFB+.22Taylor et al.26@ pointed out that by using progressively larger basis sets along with "singles and doubles" CI makes the agreement between the calculated and experimental geometry of HCN progressively worse. The answer to the problem according to Taylor et al.25926is not in extending the basis set but in the inclusion of multiple excitations as evidenced by the importance of unlinked clusters. The purpose of this investigation was to determine a variety of one-electron properties at the equilibrium geometry for the isoelectronic molecules being studied. These calculated properties will be compared and contrasted with available experimental and theoretical results. Method and Results Calculations were carried out on an IBM 370-158 computer by using double l quality basis sets (13S, 7P/4S, 2P) for boron, carbon, nitrogen, oxygen, and fluorine and (6S/2S) for hydrogen, optimized by Huzinaga and Sakaiqn The IBMOL5A ab initio program package developed by Clementi et a1.2 was utilized. The one-electron property part of the package is essentially identical with that employed in the POLYATOM/~ program.' The geometries were optimized by using a 0.005-8 grid and optimizations were repeated for each parameter till self-consistency was achieved. The points on the energy surfaces showed no evidence of random errors giving smooth potential surfaces. The minima were graphically determined, the uncertainty in the geometrical parameters being less than 0.001 A. Table I gives the equilibrium geometries and total energies for all of the systems studied. Also included are the virial theorem results for these optimized systems. Some one-electron properties, relative to the center of mass of the respective molecules, are given in Table 11, all properties being given in atomic units with the appropriate conversion factors given in Table 111. The coordinate system used in all cases has the 2 axis in the direction Y H. Positive signs on the dipole moment indicate H+XYpolarity. Atom-centered one-electron properties are given in Table IV. Again all properties are given in atomic units and reference should be made to Table I11 for the conversion factors. Table V lists the various quadrupole +
2908
The Journal of Physical Chemistry, Vol. 83, No. 22, 1979
Tyrrell
TABLE 11: Center-of-Mass One-Electron Properties for HXYa potential electronic total electric field electronic total field gradient electronic total dipole moment electronic total quadrupole moment electronic total second moment electronic total octupole moment electronic total av diam susceptibility a
HCN
HNC
HBO
HOB
HBF+
HFB+
-10.5315 1.7959
-10.7841 1.9762
-10.4730 1.5222
-10.9004 1.a099
-11.0796 1.9789
-8.7294 0.6814
-0.3780 -0.3408
2.1336 -2.9299
-2.5655 1.1966
4.2059 -4.0441
-4.2698 2.2780
3.9304 -0.9056
7.8573 -12.7197
8.5748 -16.0501
7.9351 -11.3487
9.0547 -17.0852
9.6676 -13.0738
5.7439 -3.6847
-0.1659 1.3454
-0.1489 1.2183
0.3035 1.3722
-0.5420 1.5471
0.4623 2.0288
-2.8556 -0.7359
-22.9904 1.5754
-21.9391 1.3447
-28.3007 -0.4514
-25.2161 -0.2851
-28.5848 2.2785
-45.6662 8.8376
-50.5186 -25.9528
-49.5548 -26.2710
-53.3575 -25.5081
-52.8980 -27.9671
-47.0482 -16.1849
-66.1189 -11.6151
6.6109 19.9838 39.53
-42.2758 -3.2858 42.57
29.4993 35.9306 42.20
-46.2732 3.3407 37.53
-16.1124 9.2121 40.30
45.4076 16.0238 52.75
All properties are in atomic units except for average diamagnetic susceptibility which is in units of cm3/mol X
TABLE 111: Atomic Unit Conversion Factors
1 au = 9.07618 esu/cm 1a u = 0.171524 x l o 8 dyn/esu 1a u = field gradient 0.324123 X 10l6esu/cm3 1 au = 2.54177 D dipole moment quadrupole moment 1au = 1.34504 X lo-'' esu.cm2 second moment 1au = cm2 0.2800285 X octupole moment 1a u = 0.711766 x lo-" esu.cm3 diamagnetic susceptibility 1 au = 1.188028 x cm3/mol diamagnetic shielding 1 au = 17.75 ppm 1 au = 627.5 kcal/rnol energy potential electric field
coupling constants obtained by using the values for the nuclear quadrupole moments given in the footnotes. Table VI compares the present results for a number of oneelectron properties of HCN with experimental and previous calculated results.
Discussion The geometries and energies given in Table I show similar patterns to those observed in a more extensive set of calculations for the same isoelectronic series with a 4-31G basis set.22 This is not unexpected since the basis set used in the present work, while much larger than the 4-31G basis set, does use a similar contraction scheme for the valence shell. In all instances the species HXY, where X is the less electronegative atom, is more stable than where X is the more electronegative of the two heavy atoms. Further, for the less stable member of each pair of molecules the highest occupied valence orbital is a nonbonding orbital associated with the less electronegative heavy atom. On the other hand, the highest occupied molecular orbitals in the more stable member of each pair are two degenerate a orbitals except in the case of HBFS. In both HBF+ and HFB" it would be more correct to describe the degenerate a orbitals as fluorine nonbonding orbitals. The bond lengths between the two heavy atoms are consistently shorter in the species where the hydrogen is bound to the less electronegative atom.
TABLE I V : Atom-Centered, One-Electron Properties for HXYa HCN HNC HBO potentia?' H X
Y electric fieldC H X Y electric field gradientC H X Y quadrupole moment H X Y av diamagnetic shielding H X Y
HOB
HBF'
HFB' -6.6291 -28.2367 -13.5441
-6.5786 -24.9314 -14.9466
-4.9242 -15.1598 -28.5167
0.0821 0.0340 -0.2129
0.0760 0.0087 -0.0774
0.0919 0.0454 -0.2658
0,0901 -0.1277 -0.0248
-0.5113 -0.1815 0.9789
-0.2614 0.4161 0.4361
-0.5944 -1.6379 0.4422
-0.2361 0.6014 -1.6082
-0.5380 -3.2824 0.0412
-5.3765 -0.8596 4.4815
-9.7804 -3.7955 2.4383
-8.4968 -3.0180 4.4371
0.5185 -1.4080 7.2452
22.5195 12.3367 11.3273
-5.6264 -18.4013 -21.3247
-6.1617 -21.5931 -18.1211
0.0708 0.0835 -0.1995
0.0725 0.0731 -0.1408
-0.3800 0.4096 1.2074 -6.6122 -1.2426 4.5796 -99.87 -326.62 -378.51
lo6.
- 109.37 -383.28 -321.65
-5.1012 -15.3464 -24.7535
-90.55 -272.40 -439.37
-116.77 -442.53 -265.30
-87.40 -269.09 -506.17
-117.67 -501.20 -240.41
a AH properties are in atomic units except average diamagnetic shielding which are in parts per million (ppm). Total contribution to electric field. electronic contribution to potential.
Only
The Journal of Physical Chemistty, Vol. 83, No. 22, 1979 2909
Electronic Structure of Isoelectronic Molecules
TABLE V : Quadrupole Coupling Constants (-eq Q / h ) a 2H
I4N 1°B l1
HFB+
HCN
HNC
HBO
HOB
0.2498 -4.4891
0.3361 0.6746
0.1718
0.3907
0.1552
0.3536
-8.4105 -4.1075 2.7315
-8.9372 -4.3647 -10.2585
-12.1560 -5.9367
-0.8333 -0.4070
B
l'0
HBF+
Units are MHz. OsH = t 0 . 0 0 2 796 5 b, O L ~t 0~. 0 = 1 5 82 b, Olog= +0.086 b, O i l g = +0.042 b,
0
1
7
=~ -0.026 6 5 b.
TABLE VI: One-Electron Properties of HCNa this work
2.9868
quadrupole moment, (esu.cma)X octupole moment, (esuem3)x 10% quadrupole coupling constant, MHz
2.12 6.56
3.1 ? 0.6"
3.26,16 3.244," 3.04,16 3.22,14 3.32519 1.93,'' 2.0319 6.37l'
0.2498 -4.489
0.194,90.199," 0.20218" -4.714'
0.208'6 -4.65
99.87 326.62 378.51
101.7 2 3' 3 2 6.7 5 3' 37 7.62 32
97 l 3 32613 381i3
I4N av diamagnetic shielding, ppm % V d W
uav:(C) uav ( N )
Superscripts indicate the appropriate reference sources.
The geometries obtained for both HCN and HNC give bond lengths which are consistently about 0.01 A shorter than the experimental values. It should be pointed out that the bond lengths calculated correspond to re values. There is excellent agreement between the present geometry for HBO and that obtained by BotschwinaZ3while both differ significantly from the results of Thomson and Wishart.zl On the other hand, the present result for the geometry of HOB has both bond lengths coming out 0.01-0.02 A shorter than the values obtained by Bots c h ~ i n a The . ~ ~ geometry for HBF+ is in good agreement with the previous 4-31G basis set resultsz2while the geometry for HFB+ emphasizes even more than the 4-31G basis set results22the picture of HFB' as essentially HF B+. The only experimental dipole moment available for the systems presently being considered is for HCN which has a value of 2.986 D8 so that our calculated value of 3.4 D is somewhat large. Many large basis set calculations of the dipole moment of HCN do overestimate its value; for example, Bonaccorsi et al.le obtained 3.26 D, Ditchfield et al.I7 obtained 3.244 D, Johansson et a1.283.04 D, and Pearson et al.I4 3.22 D. On the other hand, minimal basis sets or semiempirical calculations significantly underestimate the dipole moment. Similarly for HNC our calculated value for the dipole moment (3.094 D) is in reasonably good agreement with Pearson et all4 (2.91 D) but somewhat larger. The dipole moment obtained by Thomson and WishartZ1for HI30 (3.261 D) is about 0.2 D smaller than our result (3.485 D). All of the calculated dipoles are in the same direction except that of HFB+ which would indicate a H-FB+ polarity. In all cases except HFB+ the nuclear contribution to the dipole dominates and determines the sign of the dipole. The contribution of the A orbitals increases in magnitude and changes sign in going from the more stable to the less stable member of each HXY pair of molecules as is also the case with the UX-y contribution. The UH-x contribution, however, decreases in magnitude in going from the more stable to the less stable member of each HXY pair and in addition retains its sign. It should be noted that in the more stable member of each pair the UH-X contribution is of the opposite sign to that of the n x - y and contributions while in the less stable member all bonding orbitals contribute in the same sense. The dipole moment is invariant to
+
other work
3.420
'H
a
experimental
dipole moment, D
choice of origin only in those species with no net charge. The quadrupole moments, OZZ, and the octupole moments, QzZz,are given relative to the center of mass of the molecule in Table 11. The only experimental value available for comparison is QZz = 3.1 f 0.6 esu.cm2 for HCl6N.lZ This can be compared with the values of 0.93 and 1.93 eswcmz obtained by using STO-5G and 4-31G basis sets, r e s p e c t i ~ e l ywith , ~ ~ the value of 2.03 eswcm2 obtained by Gready et al.19 and the present value of 2.12 eswcmz. The quadrupole and octupole moments for HCN and HNC are in good agreement with the values obtained by Gready et al.19 Pan and Allen30 calculated the quadrupole moment at the carbon in HCN to be 1.624 X eswcm2which compares well with the value of 1.671 X lo-% esu.cmz obtained in the current investigation. The atom-centered quadrupole moments for HBO should be compared with those of Thomson and Wishartzl which are of the same sign and of similar relative magnitude but approximately a factor of 2 larger. This discrepancy may be due to the method of definition of the quadrupole moment. The definition used in this work is that due to Buckingham31 but some workers use the definition given by Townes and Schaw10w~~ which increases the moment by a factor of 2. The nuclear and electronic contributions to the quadrupole moments are similar in magnitude but of opposite sign with the nuclear contributions dominating in HCN, HNC, HBF+, and HFB+ while the reverse is true for HBO and HOB. All molecular orbital contributions to the quadrupole moments are of the same sign with the dominant ones being due to the outermost nonbonding orbital and the u ~ orbital. - ~ The nuclear contributions to the octupole moments are positive except in the case of HFB+ and are in every case much larger for the more stable member of each pair of HXY molecules, The electronic contributions, on the other hand, alternate in sign, being negative for the more stable member of the pair. The dominant molecular orbital contributions are due to the highest nonbonding orbital and the uH-X orbital. These contributions retain their sign in going from one member of each pair to the other while the remaining bonding orbital and nonvalence orbital contributions switch signs. The field gradients measured relative to the atom centers can be used to determine quadrupole coupling constants, -eqQ/h, if the appropriate atomic quadrupole moment, 8, is known. Pan and Allen30 determined the
2910
The Journal of Physical Chemistry, Vol. 83, No. 22, 1979
field gradient at the hydrogen in DCN and at the carbon in H14CN to be 0.472 and -0.481 au, respectively, which should be compared to the values obtained in the present calculation of -0.380 and 0.410 au, respectively. The sign difference is due to Pan and Allen having the positive direction of the 2 axis pointing toward the nitrogen. The only experimental values for the quadrupole coupling constants are those for 2H in 2HCN9-11and for 14N in HC14N8and the present values of 249.8 kHz and -4.489 MHz, respectively, should be compared with these and with the calculated values of 208 kHz and -4.65 MHz, respectively, obtained by Bonaccorsi et al.16 Particularly noticeable are the large changes in the quadrupole coupling constants at the various nuclei for the isoelectronic pairs of molecules. This is especially noticeable for 14Nin HCN and HNC and for 170in HBO and HOB and is attributable primarily to large changes in the electronic contribution to the field gradient at these centers. On the other hand, the boron quadrupole coupling constants in HBO and HOB show only a small variation and in this case the electronic contribution to the field gradient at the boron in these two molecules shows little variation. For an exact Hartree-Fock wave function the Hellman-Feynman theorem predicts that the sum of the forces on all the nuclei should be zero for all internuclear dist a n c e ~ The . ~ ~ net force on each nucleus should be zero for the equilibrium geometry determined by using an approximate Hartree-Fock wave function which is fully optimized. This net force is the product of the electric field at the nucleus times the charge on the nucleus. This corresponds to total forces at the nuclei ranging from 1.708 X dyn in the negative 2 direction for HBF’ to 2.147 X dyn in the same direction for HNC so the Hellman-Feynman is not in fact satisfied. Presumably these deviations from the Hellman-Feynman theorem are due, at least in part, to incompletely optimized wave functions. I t is interesting that the total forces at the nuclei are smaller for the less stable member of each HXY pair. The average diamagnetic susceptibility, ,:,x of the molecule can be determined from the electronic contribution to the second moment of the charge distribution relative to the center-of-mass with the relation xa: = (-e2N/6mc2)( r2)CM The results are shown in Table 11. The sign of these quantities is positive because of the choice of axes directions. Unfortunately no experimentaldata are available for comparison, the only measured quantity being xl - x = 7.2 f 0.4 X lo4 cm3 mol. The calculated value of xavd obtained by Kat0 et of -14.87 X lo4 cm3/molfor HCN is more than a factor of 2 smaller than the present result. The electronic contribution to the potential at a specific atom in a molecule can be used to determine the average diamagnetic shielding at that atom by using the relation a,,d(A) = (e2/3mc3)(l / r > A where a,:(A) is the average diamagnetic shielding at atom A in parts per million (ppm). The results are given in Table IV. The values for HCN are in excellent agreement with the experimental results obtained by Garvey and DeLuciaS4for a,d(lH), aa$(13C), and a,$(””) of 101.72, 326.75, and 377.62 ppm, respectively, and with the results of Gierke and F1~gare.l~ The significant changes in ua: at a particular atom in going from one molecule to another are worth noting. Generally, theories of chemical shifts in molecules assume that they arise primarily from changes in paramagnetic terms. The present results are in agreement with other recent calculations20in suggesting that diamagnetic effects must also be taken into account.
al
Tyrrell
In conclusion it does appear that the calculated oneelectron properties, where they can be compared with experimental results, are in quite good agreement with these results. On the basis of comparison with experimental data, where available, the present calculated bond lengths are too short by approximately 0.01 A and the dipole moments are too large by about 15%. While the inclusion of polarization functions has generally been found to lead to improvements in the one-electron properties calculated at a specific geometry, somewhat the reverse effect has been observed in geometry optimization of bond lengths. One would therefore expect that use of a double { plus polarization basis set would give better one-electron properties for the particular equilibrium geometry obtained but that equilibrium geometry would not agree as well with the experimental geometry as the one obtained by using the double { basis set. The error in the calculated dipole moments is at least in part due to the insufficiency of the present basis sets in describing the behavior of electrons distant from the nuclei. It is difficult to determine how accurately other multipole moments are calculated because of the lack of experimental data but the present results are in good agreement with previous calculations, where available. The discrepancy between the present calculated value of the deuteron quadrupole coupling constant for DCN and the experimental and calculated values (Table VI) is directly attributable to the sensitivity of this quantity to the D-C bond distance. Bonaccorsi’sla calculated value was obtained at a D-C bond distance of 1.065 A compared to the 1.056 A distance in the present calculation. Those properties which are less sensitive to geometry or electron distribution far from the nuclei, such as the 14N quadrupole coupling constant and average diamagnetic shielding, are in good agreement with experiment where such comparison is possible. References and Notes (1) D. B. Neumann, H. Basch, R. L. Kornegay, L. C. Snyder, J. W. Moskowitz, C. Hornback, and S. P. Liebmann, Quantum Chemistry Program Exchange No. 199, The Polyatom (Verslon 2) System of Programs for Quantitative Theoremical chemistry. (2) E. Clementi, J. Mehl, and H. Popkle, IBMOLBA Users Guide, IBM Research Laboratory, San Jose, Callf. 95193. (3) L. C. Snyder and H. Basch, “Molecular Wave Functions and Properties”, Wiiey-Intersclence, New York, 1972. (4) R. F. Porter and E. R. Long, J . Am. Chem. Soc., 93, 6301 (1971). (5) C. C. Costain, J . Cbem. Phys., 29, 884 (1958). (6) R. A. Creswell and A. G. Robiette, Mol. Pbys., 38, 869 (1978). (7) E. F. Pearson, R. A. Creswell, M. Wlnnewlsser, and G. Winnewisser, 2. Naturforscb A , 31, 1394 (1976). (8) B. N. Bhattacharya and W. Gordy, Pbys. Rev., 119, 144 (1960). (9) F. C. DeLucia and W. Gordy, Pbys. Rev., 187, 58 (1969). (IO) W. J. Caspary, F. Mlllett, M. Reichbach, and B. P. Dailey, J . Cbem. Pbys., 51, 623 (1969). (11) G. Tomasevich, quoted in S. C. Wofsy, J. S. Muenter, and W. KiemDerer. J. Cbem. Pbys.. 53, 4005 (1970). S. L.’Hartford, W. C. Allen, C. L. Norris,‘E. F. Pearson, and W. H. Flygare, Cbem. Phys. Letts., 18, 153 (1973). T. D. Geke and W. H. Ftygare, J. Am. Cbem. Sm., 94, 7277 (1972). P. K. Pearson, 0. L. Biackman, H. F. Schaefer, B. Roos, and U. Wahigren, Astropbys. J., 184, L19 (1973). C. W. Kern and M. Karplus, J. Cbem. Phys., 42, 1062 (1965). R. Bonaccorsi, E. Scrocco, and J. Tomasi, J. Cbem. Pbys., 50, 2940 (1969). R. Ditchfield, D. P. Mlller, and J. A. Pople, J. Cbem. Pbys., 54, 4186 (1971). Y. Kato, Y. Fujimoto, and A. Saika, Cbem. phys. Left., 13, 453 (1972). J. E. Oready, G. B. Bacskay, and N. S. Hush, Cbem. Pbys., 31, 467 (1978). M. Barfield and D. M. Grant, J . Cbem. Pbys., 67, 3322 (1977). C. Thomson and B. J. Wlshart, Tbeor. Cbim. Acta, 35, 267 (1974). N. L. Summers and J. Tyrrell, J. Am. Cbem. Soc., 99, 3960 (1977). P. Botschwlna, Cbem. Pbys., 28, 231 (1978). H. B. Jansen and B. Ros, Tbeor. Cbim. Acta, 21, 199 (1971). P. R. Taylor, G. B. Bacskay, N. S. Hush, and A. C. Hurley, J . Cbem. Pbys., 89, 1971 (1978). P. R. Taylor, G. B. Bacskay, N. S. Hush, and A. C. Hurley, J. Cbem. Pbys., 89, 4669 (1978). S. Huzinaga and Y. Sakai, J. Cbem. Pbys., 50, 1371 (1969).
The Journal of Physical Chemistry, Vol. 83, No. 22, 1979 2911
Dielectric Saturation Effects (28) A. Johansson, P. Kollman, and S.Rothenberg, Chem. Phys. Left., 18, 123 (1972). (29) R. Ditchfield, D. P. Miller, and J. A. Pople, quoted by R. Dltchfield in “Critical Evaluationof Chemical and Physical Structural Information”, D. R. Lide and M. A. Paul, Ed., National Academy of Sciences, Washington, D.C., 1974.
(30) D. C. Pan and L. C. Allen, J. Chem. Phys., 46, 1797 (1967). (31) A. D. Bucklngham, 0.Rev. Chem. SOC., 13, 183 (1959). (32) C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy”, McGraw-Hili, New York, 1955. (33) C. W. Kern and M. Karplus, J . Chem. Phys., 40, 1374 (1964). (34) R. M. Garvey and F. C. DeLucia, J. Mol. Spectrosc., 50, 38 (1974).
Effects of Dielectric Saturation on Planar Double Layer Interactions for No Added Salt Hiroshi Maeda” and Fumlo Oosawat Department of Chemistry, Faculty of Science, Nagoya University, Nagoya, 464, Japan (Received October 10, 1978; Revised Manuscript Received February 12, 1979) Publication costs assisted by Nagoya University
The effect of dielectric saturation of water on the interaction between electric double layers on two charged plates is theoretically investigated in the absence of low molecular weight salts. The dielectric constant of water, e , is given as a function of the electric displacement, D, instead of the electric field, E, by e = (to + 3aD2)/(1 aD2)or in shorter form by = eo/(l + aD2),which is derived on the basis of the assumption that the self-free energy of polarized dielectrics is expressed as (up2+ bP)where P is the dielectric polarization. By using this expression for e, we solved the Poisson-Boltzmann equation analytically,and the potential, the field, and the counterion concentration were obtained as functions of the distance from the plates for various values of the charge density on the plates. The electric energy, entropy, and free energy are also calculated, and the effect of dielectric saturation is estimated for various values of the parameter a. The counterion activity cannot exceed a certain limit as the charge density increases and the limit decreases with increasing distance between the two plates. The electric field is made small by the dielectric saturation except in the vicinity of the plates where the field is made large. The electric entropy is decreased by the dielectric saturation (its absolute value is increased),while the electric energy is little influenced. The electric displacement is made smaller everywhere except at the surface of the plate. The effect of dielectric saturation becomes less significant with increasing distance between the two plates.
+
Introduction The effect of dielectric saturation on the interaction between planar electric doouble layers in low molecular weight salts has been examined theoretically by several a~thors.l-~ Both Sparnaay2and Levine and Bell3expressed the dielectric constant of water, t, in a power series of the electric field, E, and assumed the form
a’E2 (1) The value of the coefficient a’ used by them was 3 X or 3 X lo-’ (cm2/esu)2,respectively. With t given by eq 1, however, the one-dimensional Poisson-Boltzmann equation cannot be solved analytically. Therefore, Sparnaay gave an approximate solution by a perturbation method and Levine and Bell solved the equation by numerical integration. For the study of the interaction between highly charged plates in water, it is important to find an expression for t which is applicable to high field strengths and convenient for analytical integration of the Poisson-Boltzmann equation. Grahamel proposed the following form of t as a function of E with only one parameter b to cover high field strengths. e = 3 + [(to - 3)b/Eb1I2] arctan (t~l/~E) (2) If the right-hand side is expanded in a power series of E, the coefficient a’ in eq 1 is found to be equivalent to (eO - 3)b/3. He pointed out that eq 2 agrees well with the e
to -
Institute of Molecular Biology, Faculty of Science, Nagoya University.
0022-365417912083-2911$01 ,0010
result of the theory of Booth,4 if b is set at 1.08 X (cm2/esu)2.Unfortunately, however, eq 2 is not convenient for integration of the Poisson-Boltzmann equation. Most of the theories on dielectric saturation have been concerned with derivation of the dielectric constant as a function of ESp7The basic idea of this work is to express the dielectric constant as a function of electric displacement, D,and to solve the Poisson-Boltzmann equation for planar double layers.
A Phenomenological Approach to the Dielectric Saturation Effect Let us consider a large dielectric which is placed between two parallel charged plates and polarized in a constant external field, Eo. The free-energy density is given by the following expression:
F = (1/47r)JD 0 E dD
(3)
This free-energy density is composed of three terms: F = Fo Fint+ F, (4) The first term, Fo, is the free-energy density in a reference state where the dielectric is removed from the external field; this is given by E 2 / 8 ~ .The second term, Fht, which is due to the interaction between the electric field and the polarized dielectric is equal t o -Ed‘, where P is the dielectric polarization. The work necessary to polarize the dielectric is called self-energy and denoted here as F,. According to Landau’s expansion: this free-energydensity F, is expressed in a power series of P as follows:
+
0 1979 American Chemical Society
